An algebraic element over a field 
in an extension field 
is an element 
that is a root
of some nonzero polynomial
with coefficients in 
.
Every element of 
is algebraic over 
. The sum, difference, product,
and quotient of algebraic elements in 
are again algebraic over 
, with the denominator of a quotient
required to be nonzero. It follows that the simple extension field 
is an algebraic
extension of 
iff 
is algebraic over 
.
The imaginary unit i is algebraic over the field 
of real numbers since it is a root of the polynomial 
. Because its coefficients are integers,
it is even true that 
is algebraic over the field 
of rational numbers, i.e., it is an algebraic
number (and also an algebraic integer).
As a consequence, 
and 
are algebraic extensions of 
and 
respectively. (Here, 
is the complex field 
, whereas 
is the total ring
of fractions of the ring of Gaussian integers ![Z[i]](/images/equations/AlgebraicElement/Inline24.svg)
.)
An algebraic simple extension field 
is always a finite
extension. Its extension field degree
is the polynomial degree of the extension
field minimal polynomial of 
over 
. More generally, any finite list of algebraic elements generates
a finite extension, which explains closure under
the arithmetic operations above (Milne 2022, pp. 18-19).
